A new class of analytic charged spherically symmetric black hole solutions, which behave asymptotically as flat or (anti-)de Sitter spacetimes, is derived for specific classes of $f(R)$ gravity, i.e., $f(R)=R-2\alpha \sqrt{R}$ and $f(R)=R-2\alpha \sqrt{R-8\mathrm{\Lambda }}$, where $\mathrm{\Lambda }$ is the cosmological constant. These black holes are characterized by the dimensional parameter $\alpha$ that makes solutions deviate from the standard solutions of general relativity. The Kretschmann scalar and squared Ricci tensor are shown to depend on the parameter $\alpha$, which is not allowed to be zero. Thermodynamical quantities, like entropy, Hawking temperature, quasilocal energy, and the Gibbs free energy are calculated. From these calculations, it is possible to put a constraint on the dimensional parameter $\alpha$ to have $0<\alpha <0.5$ so that all thermodynamical quantities have a physical meaning. The interesting result of these calculations is the possibility of a negative black hole entropy. Furthermore, present calculations show that for negative energy particles inside a black hole,behave as if they have a negative entropy. This fact gives rise to instability for $f_{RR}<0$. Finally, we study the linear metric perturbations of the derived black hole solution. We show that for the odd-type modes our black hole is always stable and has a radial speed with fixed value equal to 1. We also use the geodesic deviation to derive further stability conditions.

We derive a new interior solution for stellar compact objects in $$f\mathcal {(R)}$$f ( R ) gravity assuming a differential relation to constrain the Ricci curvature scalar. To this aim, we consider specific forms for the radial component of the metric and the first derivative of $$f\mathcal {(R)}$$f ( R ) . After, the time component of the metric potential and the form of $$f({\mathcal {R}})$$f ( R ) function are derived. From these results, it is possible to obtain the radial and tangential components of pressure and the density. The resulting interior solution represents a physically motivated anisotropic neutron star model. It is possible to match it with a boundary exterior solution. From this matching, the components of metric potentials can be rewritten in terms of a compactness parameter C which has to be $$C=2GM/Rc^2<<0.5$$C = 2 G M / R c 2 < < 0.5 for physical consistency. Other physical conditions for real stellar objects are taken into account according to the solution. We show that the model accurately bypasses conditions like the finiteness of radial and tangential pressures, and energy density at the center of the star, the positivity of these components through the stellar structure, and the negativity of the gradients. These conditions are satisfied if the energy-conditions hold. Moreover, we study the stability of the model by showing that Tolman–Oppenheimer–Volkoff equation is at hydrostatic equilibrium. The solution is matched with observational data of millisecond pulsars with a withe dwarf companion and pulsars presenting thermonuclear bursts.

We investigate a bounce inflation model with a graceful exit into the Friedmann-Robertson-Walker (FRW) decelerated Universe within $f(T)$-gravity framework, where $T$ is the torsion scalar in the teleparallelism. We study the cosmic thermal evolution, the model predicts a supercold universe during the precontraction phase, which is consistent with the requirements of the slow-roll models, while it performs a reheating period by the end of the contraction with a maximum temperature just below the grand unified theory (GUT) temperature. However, it matches the radiation temperature of the hot big bang at later stages. The equation-of-state due to the effective gravitational sector suggests that our Universe is self-accelerated by teleparallel gravity. We assume the matter component to be a canonical scalar field. We obtain the scalar field potential that is induced by the $f(T)$ theory. The power spectrum of the model is nearly scale invariant. In addition, we show that the model unifies inflaton and quintessence fields in a single model. Also, we revisited the primordial fluctuations in $f(T)$ bounce cosmology, to study the fluctuations that are produced at the precontraction phase.

A tetrad field with spherical symmetry is applied to the charged field equations of $f(T)$ gravity theory. A special spherically symmetric charged-dS solution is obtained. The scalar torsion of this solution is a vanishing quantity. The spacetime of the derived solution is rewritten as a multiplication of three matrices: The first matrix is a special case of Euler's angle ``so(3),'' the second matrix represents a boost transformation, while the third matrix is the square root of the spherically symmetric charged-dS metric. It is shown that the boost matrix is important because it plays an essential role in adjusting the spacetime to become a solution for $f(T)$ theory.